Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2010, Article ID 987372, 11 pages
doi:10.1155/2010/987372
Research Article
Global Exponential Stability of
Impulsive Functional Differential Equations
via Razumikhin Technique
Shiguo Peng1 and Liping Yang2
1
2
College of Automation, Guangdong University of Technology, Guangzhou 510006, China
College of Applied Mathematics, Guangdong University of Technology, Guangzhou 510006, China
Correspondence should be addressed to Shiguo Peng, psg7202@126.com
Received 1 November 2009; Accepted 23 January 2010
Academic Editor: Yong Zhou
Copyright q 2010 S. Peng and L. Yang. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
This paper develops some new Razumikhin-type theorems on global exponential stability of
impulsive functional differential equations. Some applications are given to impulsive delay
differential equations. Compared with some existing works, a distinctive feature of this paper
is to address exponential stability problems for any finite delay. It is shown that the functional
differential equations can be globally exponentially stabilized by impulses even if it may be
unstable itself. Two examples verify the effectiveness of the proposed results.
1. Introduction
Functional differential equations FDEs which include delay differential equations DDEs
play a very important role in formulation and analysis in mechanical, electrical, control
engineering and physical sciences, economic, and social sciences 1, 2. Therefore, the theory
of FDEs has been developed very quickly. The investigation for FDEs has attracted the
considerable attention of researchers and many qualitative theories of FDEs have been
obtained. A large number of stability criteria of FDEs have been reported.
In addition to the delay effect, as is well known, impulsive effect is likely to exist in
a wide variety of evolutionary processes in which states are changed abruptly at certain
moments of time in the fields such as medicine and biology, economics, electronics, and
telecommunications 3. So far, a lot of interesting results on stability have been reported
that have focused on the impulsive effect of FDEs see, e.g., 4–16 and the references cited
therein.
2
Abstract and Applied Analysis
In particular, several papers devoted to the study of exponential stability of impulsive
functional differential equations IFDEs have appeared during the past years. In 14, 15, the
authors have investigated exponential stability of IFDEs by using the method of Lyapunov
functions and Razumikhin techniques. In 16, the authors have also studied exponential
stability by using the method of Lyapunov functional. However, some results in 15, 16
imposed a restrictive condition on time delays which were less than the length of all the
impulsive intervals see, e.g., 15, Theorems 3.1-3.2 and 16, Theorem 3.1. The aim of
this paper is to establish global exponential stability criteria for IFDEs by employing the
Razumikhin technique which illustrate that impulses do contribute to the stability of some
IFDEs and the restrictive condition that the time delays are less than the length of all the
impulsive intervals can be removed in this paper.
2. Preliminaries
Throughout this paper, unless otherwise specified, we use the following notations. Let R −∞, ∞, R 0, ∞, N {1, 2, . . .}, I be the identity matrix, λmax · and λmin · be the
maximal eigenvalue and the minimal eigenvalue of a matrix, respectively.√If A is a vector or
matrix, its transpose is denoted by AT . For x ∈ Rn and A ∈ Rn×n , let |x| xT x be Euclidean
vector norm, and denote the induced matrix norm by
|Ax| λmax AT A .
x/
0 |x|
A sup
2.1
Let τ > 0 and C C−τ, 0; Rn denote the family of all bounded continuous Rn valued functions φ defined on −τ, 0. P CI; Rn {ψ : I → Rn | ψs is continuous for
all but at most countable points s ∈ I and at these points s ∈ I, ψs and ψs− exist and
ψs ψs}, where I ⊂ R is an interval, ψs and ψs− denote the right-hand and lefthand limits of the function ψs at time s, respectively. Especially, let P C P C−τ, 0; Rn with norm ψ sup−τs0 |ψs|.
In this paper, we consider the following IFDEs:
x t ft, xt , t / tk , t 0,
−
− Δxtk x tk − x tk Ik x tk , t tk , k ∈ N,
xs φs,
2.2
−τ s 0,
where f : R × P C → Rn , xt θ xt θ, θ ∈ −τ, 0. The initial function φ ∈ P C. The
impulsive function Ik ∈ CRn ; Rn k ∈ N, and the impulsive moments tk k 1, 2, . . .
satisfy 0 t0 < t1 < t2 < · · · , and limk → ∞ tk ∞.
In this paper, we assume that functions f and Ik , k ∈ N, satisfy all necessary conditions
for the global existence and uniqueness of solutions for all t t0 . Denote by xt xt, t0 , φ
the solution of 2.2 such that xt0 φ. For the purpose of stability in this paper, we also
assume that ft, 0 0 and Ik 0 0, k ∈ N. So system 2.2 admits a zero solution or trivial
solution xt, t0 , 0 0. We further assume that all the solutions xt of 2.2 are continuous
except at tk , k ∈ N, at which xt is right continuous, that is, xtk xtk , k ∈ N.
Abstract and Applied Analysis
3
Definition 2.1. The trivial solution of system 2.2 is said to be globally exponentially stable, if
there exist numbers λ > 0 and M 1 such that
x t, t0 , φ Mφe−λt ,
t 0,
2.3
whenever φ ∈ P C.
Definition 2.2. V : −τ, ∞×Rn → R is said to belong to the class ν0 , if V is continuous on each
of the sets tk−1 , tk × Rn , limt,y → t−k ,x V t, y V t−k , x exists, V t, x is locally Lipschitzian
in all x ∈ Rn , and V t, 0 ≡ 0 for all t ≥ −τ.
Definition 2.3. V : −τ, ∞ × Rn → R is said to belong to the class ν1 , if V is continuous on
each of the sets tk−1 , tk × Rn , V t, 0 ≡ 0 for all t ≥ −τ, limt,y → t−k ,x V t, y V t−k , x exists,
Vt t, x, Vx t, x are continuous, where t, x ∈ tk−1 , tk × Rn , k ∈ N,
Vt t, x ∂V t, x
,
∂t
Vx t, x ∂V t, x
∂V t, x
,...,
.
∂x1
∂xn
2.4
Definition 2.4. Given a function V : −τ, ∞ × Rn → R , the upper right-hand derivative of
V with respect to system 2.2 is defined by
1 D V t, ϕ0 lim sup V t h, ϕ0 hF t, ϕ − V t, ϕ0
h → 0 h
2.5
for t, ϕ ∈ R × P C.
3. Razumikhin-Type Theorems
In this section, we will present some Razumikhin-type theorems on global exponential
stability for system 2.2 based on the Lyapunov-Razumikhin method.
Theorem 3.1. Let ρ supk∈N {tk − tk−1 } < ∞ and c1 , c2 , p, q, γ all be positive numbers, c a real
number, q > 1/γ > 1, and c < ln1/γ/ρ. Suppose that there exists a function V ∈ ν0 such that
i c1 |x|p V t, x c2 |x|p for all t t0 − τ, and x ∈ Rn ;
ii D V t, ϕ0 cV t, ϕ0 for all t ∈ tk−1 , tk , k ∈ N, whenever qV t, ϕ0 V t θ, ϕθ for all θ ∈ −τ, 0;
iii V tk , x Ik x γV t−k , x, k ∈ N, x ∈ Rn .
Then the trivial solution of system 2.2 is globally exponentially stable.
Proof. For any φ ∈ P C, we denote the solution xt, t0 , φ of 2.2 by xt. Without loss of
generality, we assume that φ /
0.
Since q > 1/γ and c < −ln γ/ρ, there exist positive numbers μ and h such that
q>
1
eμτ
> ,
γ
γ
Set γ1 − lnγ h/ρ, then γ < e−γ1 ρ < 1.
c <cμ<−
ln γ h
.
ρ
3.1
4
Abstract and Applied Analysis
Set Wt eμt V t, xt, we have
D Wt μWt eμt D V t, xt,
t ∈ tk−1 , tk , k ∈ N.
3.2
Let M > c2 /c1 γ be a fixed number. In the following, we will prove that
p
Wt < c1 Mφ ,
t t0 − τ.
3.3
t ∈ t0 − τ, t1 .
3.4
We first prove that
p
Wt < c1 Mφ ,
It is noted that Wt0 θ c2 φp < γc1 Mφp < c1 Mφp , θ ∈ −τ, 0. So, it only needs to
prove that Wt < c1 Mφp for t ∈ t0 , t1 . On the contrary, there exist some t ∈ t0 , t1 such
that Wt c1 Mφp . Set t∗ inf{t ∈ t0 , t1 : Wt c1 Mφp }, then we have t∗ ∈ t0 , t1 and Wt∗ c1 Mφp . Set t sup{t ∈ t0 , t∗ : Wt γc1 Mφp }. Then t ∈ t0 , t∗ and
Wt γc1 Mφp . For t ∈ t, t∗ , we have
p
Wt γc1 Mφ γWt θ,
∀θ ∈ −τ, 0.
3.5
Hence
V t, xt γe−μτ V t θ, xt θ 1
V t θ, xt θ,
q
∀θ ∈ −τ, 0.
3.6
By condition ii and 3.2, it follows that for t ∈ t, t∗ D Wt μWt eμt D V t, xt μ c Wt γ1 Wt.
3.7
∗
So, we obtain Wt∗ Wteγ1 t −t γc1 Mφp eγ1 ρ < c1 Mφp . This is a contradiction, so
3.4 holds.
Now, we assume that for some m ∈ N,
p
Wt < c1 Mφ ,
t ∈ t0 − τ, tm .
3.8
p
Wt < c1 Mφ ,
t ∈ tm , tm1 .
3.9
We will prove that
Abstract and Applied Analysis
5
Suppose not, there exist some t ∈ tm , tm1 such that Wt c1 Mφp . Set t∗ inf{t ∈
tm , tm1 : Wt c1 Mφp }. From condition iii and 3.8, we have Wtm γWt−m γc1 Mφp < c1 Mφp . Hence t∗ ∈ tm , tm1 and Wt∗ c1 Mφp . Set t sup{t ∈
tm , t∗ : Wt γc1 Mφp }. Then we have Wt γc1 Mφp . Furthermore, we have
Wt γc1 Mφp γWtθ for t ∈ t, t∗ . Thus, 3.7 holds. Then Wt∗ γc1 Mφp eγ1 ρ <
c1 Mφp , which yields a contradiction. Therefore, 3.9 holds.
By mathematical induction, we have
p
Wt < c1 Mφ ,
t 0.
3.10
|xt| < Mφe−λt ,
t 0,
3.11
Hence
where M M
1/p
, λ μ/p. The proof is therefore complete.
Theorem 3.2. Let infk∈N {tk − tk−1 } > 0 and c1 , c2 , p, q, γ, c all be positive numbers, q > γ 1,
and c > ln γ/. Suppose that there exists a function V ∈ ν0 such that
i c1 |x|p V t, x c2 |x|p for all t t0 − τ, and x ∈ Rn ,
ii D V t, ϕ0 −cV t, ϕ0 for all t ∈ tk−1 , tk k ∈ N, whenever qV t, ϕ0 V t θ, ϕθ for all θ ∈ −τ, 0;
iii V tk , x Ik x γV t−k , x, k ∈ N, x ∈ Rn .
Then the trivial solution of system 2.2 is globally exponentially stable.
Proof. Since q > γ and c > ln γ/, there exist positive numbers μ and h such that
q > γ h eμτ > γ,
ln γ 2h
.
c >c−μ>
3.12
Set q γ h and γ2 lnγ 2h/, then 1 γ < q < eγ2 . Set Wt eμt V t, xt, where xt
is defined as in the proof of Theorem 3.1.
Now, let M > qc2 /c1 , we will prove that 3.3 holds.
We first prove that 3.4 holds. In fact, it is noticed that Wt0 θ c2 φp <
1/qc1 Mφp < c1 Mφp , for θ ∈ −τ, 0. So it only needs to prove Wt < c1 Mφp
for t ∈ t0 , t1 . On the contrary, there exists t ∈ t0 , t1 such that Wt c1 Mφp . Set
t∗ inf{t ∈ t0 , t1 : Wt c1 Mφp }, then we have t∗ ∈ t0 , t1 . Set t sup{t ∈ t0 , t∗ :
Wt 1/qc1 Mφp }. Then t ∈ t0 , t∗ and Wt 1/qc1 Mφp . For t ∈ t, t∗ , we have
p
qWt c1 Mφ Wt θ,
∀θ ∈ −τ, 0.
3.13
Hence
V t, xt 1
1 −μτ
e V t θ, xt θ V t θ, xt θ,
q
q
∀θ ∈ −τ, 0.
3.14
6
Abstract and Applied Analysis
It follows that for t ∈ t, t∗ D Wt μ − c Wt −γ2 Wt,
3.15
which leads to Wt∗ Wt. This is a contradiction to the fact Wt∗ c1 Mφp >
1/qc1 Mφp Wt, so 3.4 holds.
Now, we assume that for some m ∈ N, 3.8 holds. We will prove that 3.9 holds. In
order to do this, we first claim that
p
1
W t−m c1 Mφ .
q
3.16
Suppose not, then we have Wt−m > 1/qc1 Mφp . There are two cases to be considered.
Case 1. Wt > 1/qc1 Mφp for all t ∈ tm−1 , tm .
By 3.8, we have qWt > c1 Mφp > Wt θ for θ ∈ −τ, 0 and t ∈ tm−1 , tm . Thus,
we get D Wt −γ2 Wt for t ∈ tm−1 , tm which leads to Wt−m Wtm−1 e−γ2 tm −tm−1 <
c1 Mφp e−γ2 < 1/qc1 Mφp . This is a contradiction.
Case 2. There is some t ∈ tm−1 , tm such that Wt 1/qc1 Mφp .
Set t sup{t ∈ tm−1 , tm : Wt 1/qc1 Mφp }. Then t ∈ tm−1 , tm and Wt 1/qc1 Mφp . Since for t ∈ t, tm , qWt c1 Mφp Wt θ, θ ∈ −τ, 0. By 3.15, we
have D Wt 0 for t ∈ t, tm , which gives Wt−m Wt 1/qc1 Mφp . This is also a
contradiction.
Hence, 3.16 holds. It follows from iii that Wtm γWt−m γ/qc1 Mφp <
c1 Mφp . Now, we assume that 3.9 is not true, set t∗ inf{t ∈ tm , tm1 : Wt c1 Mφp }.
Hence t∗ ∈ tm , tm1 and Wt∗ c1 Mφp . If Wt > 1/qc1 Mφp for t ∈ tm , t∗ , set
t tm , otherwise, set t sup{t ∈ tm , t∗ : Wt 1/qc1 Mφp }. Thus, we have qWt c1 Mφp Wt θ for t ∈ t, t∗ . Hence, by 3.15, D Wt 0 for t ∈ t, t∗ . Then Wt∗ Wt < c1 Mφp , which yields a contradiction. Therefore, 3.9 holds. The rest is the same as
in the proof of Theorem 3.1.
Remark 3.3. By Theorems 3.1 and 3.2, we can design impulsive control {Ik xt−k , k ∈ N} for
the following FDEs
x t ft, xt ,
xt0 s φs,
t t0 ,
−τ s 0, φ ∈ C,
3.17
such that the system can be impulsively stabilized to its trivial solution. In Theorem 3.1,
the constant c may be chosen as a positive number. In the stability theory of FDEs, the
condition D V t, ϕ0 cV t, ϕ0 allows the derivative of the Lyapunov function to be
positive which may not even guarantee the stability of functional differential system see, e.g.,
4, 15. However, as we can see from Theorem 3.1, impulses play an important role in making
a functional differential system globally exponentially stable even if it may be unstable
itself.
Abstract and Applied Analysis
7
Remark 3.4. It is important to emphasize that, in contrast with some existing exponential
stability results for IFDEs in the literature 15, 16, Theorems 3.1 and 3.2 are also valid for
any finite delay. Therefore, our new results are more practically applicable than those in the
literature, since the restrictive condition that the supper bound of time delay is less than the
length of all the impulsive intervals is actually removed here.
4. Applications and Examples
Now, we will apply the general Razumikhin-type theorems established in Section 3 to deal
with the global exponential stability of impulsive delay differential equations IDDEs.
Consider a delay system of the form
tk , t 0,
x t Ft, xt, xt − δ1 t, . . . , xt − δm t, t /
−
− Δxtk x tk − x tk Ik x tk , t tk , k ∈ N,
xs φs,
4.1
−τ s 0,
where φ ∈ P C, δi : R → 0, τ, 1 i m, are all continuous, and
F : R × Rn × Rn×m −→ Rn
4.2
is continuous. We also assume that 4.1 has a global solution which is again denoted by
xt xt, t0 , φ, Ft, 0, . . . , 0 0 and Ik 0 0, k ∈ N.
Theorem 4.1. Let ρ supk∈N {tk − tk−1 } < ∞, λ0 be a real number, and λ1 , . . . , λm , c1 , c2 , p, γ be all
positive numbers, 0 < γ < 1. Suppose that there exists a function V ∈ ν1 such that
i c1 |x|p V t, x c2 |x|p for all t t0 −τ, and x ∈ Rn ;
ii
m
Vt t, x Vx t, xF t, x, y λ0 V t, x λi V t − δi t, yi
4.3
i1
for all t ∈ tk−1 , tk k ∈ N, x ∈ Rn , y y1 , . . . , ym ∈ Rn×m ;
iii V tk , x Ik x γV t−k , x, k ∈ N, x ∈ Rn .
If λ0 m
i1 λi /γ ln γ/ρ < 0, then the trivial solution of 4.1 is globally exponentially stable.
Proof. For t ∈ tk−1 , tk , k ∈ N, φ ∈ P C, let
f t, φ F t, φ0, φ−δ1 t, . . . , φ−δm t .
4.4
Then system 4.1 becomes system 2.2, and D V t, φ0 becomes
D V t, φ0 Vt t, φ0 Vx t, φ0 F t, φ0, φ−δ1 t, . . . , φ−δm t .
4.5
8
Abstract and Applied Analysis
If λ0 m
i1
λi /γ ln γ/ρ < 0, then there exists a constant q > 1/γ, such that
λ0 q
m
ln γ
< 0.
λi ρ
i1
4.6
So, if t ∈ tk−1 , tk k ∈ N and qV t, ϕ0 V t θ, ϕθ for all θ ∈ −τ, 0, then
m
D V t, ϕ0 λ0 V t, ϕ0 λi V t − δi t, ϕ−δi t
λ0 q
m
i1
4.7
λi V t, ϕ0 cV t, ϕ0 ,
i1
where c λ0 q m
i1 λi < − ln γ/ρ. By Theorem 3.1, the trivial solution of 4.1 is globally
exponentially stable.
Corollary 4.2. Let ρ supk∈N {tk − tk−1 } < ∞. Assume that there exist scalar numbers αi 0 1 i m, 0 < α < 1 and η such that
xT Ft, x, 0 η|x|2 ,
4.8
m
F t, x, y − Ft, x, 0 αi yi ,
4.9
i1
|x Ik x| α|x|
4.10
for all t t0 , x ∈ Rn , y y1 , . . . , ym ∈ Rn×m , k ∈ N.
If
η
m
i1
α
αi
ln α
< 0,
ρ
4.11
then the trivial solution of 4.1 is globally exponentially stable.
Proof. Let V t, x xT x |x|2 , then we can easily see that condition i of Theorem 4.1 holds.
For x ∈ Rn and y y1 , . . . , ym ∈ Rn×m , from 4.8 and 4.9, we can calculate that
Vt t, x Vx t, xF t, x, y 2xT F t, x, y 2xT Ft, x, 0 2xT F t, x, y − Ft, x, 0
m
2η|x|2 2 αi |x| · yi .
4.12
i1
Let γ α2 , by the inequality ab a2 b2 /2, we have
−1/4 1/4 γ −1/2 |x|2 γ 1/2 yi 2
.
|x| · yi γ
|x| γ yi 2
4.13
Abstract and Applied Analysis
9
Substituting 4.13 into 4.12, we obtain
Vt t, x Vx t, xF t, x, y 2η m
αi
i1
|x|2 γ 1/2
α
m
m
2
2
αi yi λ0 |x|2 λi yi ,
i1
i1
4.14
1/2
αi , i 1, . . . , m.
where λ0 2η m
i1 αi /α, λi γ
From 4.11, we have
λ0 m
i1
λi
γ
ln γ
< 0.
ρ
4.15
The conclusion follows from Theorem 4.1 immediately and the proof is completed.
Remark 4.3. Let infk∈N {tk − tk−1 } > 0. If γ 1, by Theorem 3.2, we can also give some
other results on global exponential stability for 4.1. For the details, we omit them here.
Example 4.4. Consider a scalar nonlinear impulsive delay differential equation
x t Ft, xt, xt − δt, t /
tk ,
−
Δxtk ck x tk , t tk , k ∈ N,
xs φs,
4.16
s ∈ −τ, 0
on t 0, where δ : R → 0, τ is a continuous function, ck ∈ R, and
1
F t, x, y bx − x3 − y cos t,
10
4.17
with x xt, y xt − δt, b > 0.
From 4.8-4.9, we can see η b, α1 1. By Corollary 4.2, if there exists a scalar
number 0 < α < 1, such that
|1 ck | α,
b
1 ln α
< 0,
α
ρ
4.18
then the trivial solution of 4.16 is global exponentially stable.
Example 4.5. Consider the following linear impulsive delay system:
x t Axt Bxt − δt, t /
tk , t 0,
−
Δxtk Ck x tk , t tk , k ∈ N,
xs φs,
−τ s 0, φ ∈ P C,
4.19
10
Abstract and Applied Analysis
where
⎤
⎡
0.1 0.2 −0.1
⎥
⎢
⎥
A⎢
⎣0.2 0.15 0.3 ⎦,
0 0.24 0.1
⎤
⎡
−0.12 0.03 0
⎥
⎢
⎥
B⎢
⎣ 0.12 −0.2 0.05 ⎦,
0
0.14 −0.1
⎤
⎡
−0.5 0
0
⎥
⎢
⎥
Ck ⎢
⎣ 0 −0.8 0 ⎦,
0
0 −0.4
4.20
δ : R → 0, τ is a continuous function. Since λmin A −0.2315 and λmax A 0.4388, we
can not use the results in 5, 14 to determine the exponential stability. From Corollary 4.2,
we can choose η λmax A AT /2 0.4409, α1 B 0.2905, α I Ck 0.6, by 4.11,
we obtain that if supk∈N {tk − tk−1 } < −α ln α/α1 αλ1 0.5521, then trivial solution of 4.19
is globally exponentially stable.
Remark 4.6. For 4.19, the authors in 15 chose δt 1/401 e−t , τ 0.05, supk∈N {tk −
tk−1 } 0.2 such that τ tk − tk−1 0.2. However, if τ > supk∈N {tk − tk−1 }, the results in
15, 16 fail to determine the global exponential stability.
5. Conclusion
In this paper, some new Razumikhin-type theorems on global exponential stability for IFDEs
are obtained by employing Lyapunov-Razumikhin technique. Some applications to IDDEs
are also given. It should be mentioned that our results may allow us to develop an effective
impulsive control strategy to stabilize an underlying delay dynamical system even if it may
be unstable in practice, which is particularly meaningful for applications in engineering and
technology. Two examples are also given to demonstrate the effectiveness of the theoretical
results.
Acknowledgments
This work was supported by the Natural Science Foundation of Guangdong Province
8351009001000002 and the National Natural Science Foundation of China 10971232.
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